{
  "id": "1604.02505",
  "version": "v1",
  "published": "2016-04-09T01:12:51.000Z",
  "updated": "2016-04-09T01:12:51.000Z",
  "title": "Flat $δ$-vectors and their Ehrhart polynomials",
  "authors": [
    "Takayuki Hibi",
    "Akiyoshi Tsuchiya"
  ],
  "comment": "7 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We call the $\\delta$-vector of an integral convex polytope of dimension $d$ flat if the $\\delta$-vector forms $(1,0,\\ldots,0,a,\\ldots,a,0,\\ldots,0)$, where $a \\geq 1$. In this paper, we give the complete characterization of possible flat $\\delta$-vectors. Moreover, for an integral convex polytope $\\mathcal{P} \\subset \\mathbb{R}^d$ of dimension $d$, we let $i(\\mathcal{P},n)=|n\\mathcal{P} \\cap \\mathbb{Z}^N|$ and $\\ i^*(\\mathcal{P},n)=|n(\\mathcal{P} \\setminus \\partial \\mathcal{P}) \\cap \\mathbb{Z}^N|.$ By this characterization, we show that for any $d \\geq 1$ and for any $k,\\ell \\geq 0$ with $k+\\ell \\leq d-1$, there exist integral convex polytopes $\\mathcal{P}$ and $\\mathcal{Q}$ of dimension $d$ such that (i) For $t=1,\\ldots,k$, we have $i(\\mathcal{P},t)=i(\\mathcal{Q},t),$ (ii) For $t=1,\\ldots,\\ell$, we have $i^*(\\mathcal{P},t)=i^*(\\mathcal{Q},t)$ and (iii) $i(\\mathcal{P},k+1) \\neq i(\\mathcal{Q},k+1)$ and $i^*(\\mathcal{P},\\ell+1)\\neq i^*(\\mathcal{Q},\\ell+1).$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-09T01:12:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B05",
      "52B20"
    ],
    "keywords": [
      "integral convex polytope",
      "ehrhart polynomials",
      "complete characterization",
      "vector forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160402505H"
    }
  }
}